A device of this type is known from EP-A-1 068 918 wherein a hollow cylindrically designed workpiece holder is placed on the likewise hollow cylindrically designed collet chuck and is clamped to the collet chuck by means of a clamping device. The clamping device comprises a plurality of clamping elements distributed around the circumference of the collet chuck in the form of clamping balls which are functionally connected to a spring-loaded annular piston disposed in an annular space of the collet chuck. When the annular piston is acted upon hydraulically or pneumatically against the spring force the clamping balls are held in corresponding grooves of the collet chuck so that they do not project over its surface. If the pressure on the annular piston is reduced, the clamping balls are pressed by the annular piston due to the effect of the spring force into a circumferential annular groove of the workpiece holder, and in this way the workpiece holder is clamped to the collet chuck. The clamping forces are transferred over the puncitformly positioned clamping balls and are thus limited in size. If greater tensile forces occur, such devices fail due to exceedance of the Hertzian stress.